A quantum-like cognitive approach to modeling human biased selection behavior

Cognitive biases of the human mind significantly influence the human decision-making process. However, they are often neglected in modeling selection behaviors and hence deemed irrational. Here, we introduce a cognitive quantum-like approach for modeling human biases by simulating society as a quantum system and using a Quantum-like Bayesian network (QBN) structure. More specifically, we take inspiration from the electric field to improve our recent entangled QBN approach to model the initial bias due to unequal probabilities in parent nodes. Entangled QBN structure is particularly suitable for modeling bias behavior due to changing the state of systems with each observation and considering every decision-maker an integral part of society rather than an isolated agent. Hence, biases caused by emotions between agents or past personal experiences are also modeled by the social entanglement concept motivated by entanglement in quantum physics. In this regard, we propose a bias potential function and a new quantum-like entanglement witness in Hilbert space to introduce a biased variant of the entangled QBN (BEQBN) model based on quantum probability. The predictive BEQBN is evaluated on two well-known empirical tasks. Results indicate the superiority of the BEQBN by achieving the first rank compared to classical BN and six QBN approaches and presenting more realistic predictions of human behaviors.


Preliminaries
Classical Bayesian Network. Real-world agents usually work together as a system rather than acting individually. Understanding the agent's relations is an unavoidable step for reasoning and decision-making effectively. Bayesian networks are powerful graphical tools for modeling relations between agents using directional arcs between parent and child nodes. This structure provides a compact probabilistic representation of the joint distribution of a set of nodes modeled by random variables ( X i ). The inference is estimated based on the Bayes' rule: The probability of child nodes are calculated using conditional probability distributions as follows 44 : Also, probabilities under uncertainty are estimated using the TPL. Let us denote the observed node by ( O ) and unobserved ones by ( U ). The inference for some query X is given by 44 : where α is a normalization factor that guarantees the Pr(X|O) adds up to 1. Because of the confluence of artificial intelligence and statistics, Bayesian Networks are becoming increasingly popular in different areas including medicine, engineering, social sciences, economics, and many more 4 . An example of a simple BN structure for a prisoner's dilemma (PD) game, the famous benchmark in the decision-making field, is presented in Fig. 1. In this game, two prisoners are separated in isolation cells. Each prisoner can choose between two options of either cooperating or defecting with another prisoner. Figure 1 presents an example of the payoff matrix for the PD game. Before the second prisoner ( B) , makes a decision, he/she is informed that the first prisoner (A) has chosen defection ( D ), cooperation ( C ), or ( B) has no information about the first prisoner's decision (Unknown). Prisoner (A) has an equal probability of defecting or cooperating. Pr(B = D) is estimated based on the empirical results shown in Table 1 and the TPL rule: As shown in Table 1, the obtained value by (4) does not match the experimental results. TPL has been violated in other experimental observations under uncertainty repeatedly 15 . Thus, despite the impressive capabilities of BNs in inferring and predicting, the violation of TPL challenges the application of this method in predicting human behavior under uncertainty. The similarities between these inconsistencies with some contradictions in classical physics, such as Young's two-slit experiment and the ability of QP to model these discrepancies, have inspired the use of quantum probability in Bayesian networks 12 . Introduction to quantum probability. Around the 1930s, two different sets of axioms were formulated for probability theory. One of them is the Kolmogorov axioms, which are the basis of CP 49 . The second one is the Von Neumann axioms, which are the basis of QP 50 . A fundamental difference between classical and quan- (2) Pr(X 1 , X 2 , . . . , X n ) = n i=1 Pr(X i parents(X i ) .   Each node is related to a prisoner. The conditional probabilities are presented in related tables in this structure. www.nature.com/scientificreports/ tum probability theory is that CP is represented by the set theory, while QP is founded upon the Hilbert vector space 51 . QP was initially used in quantum computing, but recently the range of its applications has expanded considerably. The equivalent of a classical bit in QIT is called the qubit. The spin of an electron is an example of a qubit. In quantum computing, two eigenstate vectors |0� = 1 0 T and |1� = 0 1 T are defined to model the up and down spin states 52 . Before measurement (under an uncertainty situation), the state of a qubit can be represented as a linear combination of |0� and |1� in a two-dimensional Hilbert space ( C 2 ) as follows: So we deal with the complex coefficients, including amplitude and the phase. A unique feature of QM is that a qubit can be simultaneously in states |0� and |1�.
However, when a measurement is done |ψ� will collapse into |0� or |1� . (c 1 ) 2 and (c 2 ) 2 are the probability of finding the system in |0� and |1� , respectively. |ψ� , which is known as the initial state vector, is formed based on our knowledge about the initial state of a system. For each event |i� ( i=0,1), a projection operator P i = |i��i| is assigned to reduce the state vector |ψ� into an eigenstate |i�.The probability of finding the system in the state |i� is obtained by Eq. (6) 52 : We can extend these concepts to higher dimensions using the tensor product of |0� or |1� 52 . For example, the state of a system including two qubits can be represented by a superposition of the following four vector states: Quantum-like Bayesian Network. Quantum-like Bayesian networks achieve a significant capacity by merging the ability of QP in dealing with uncertainty and powerful features of Bayesian networks in certain situations. In 1995, Tucci 53 applied quantum probability in the BN structure for solving a problem in the physical domain. In their approach, each probability function Pr n in the real space is replaced by a complex wave function ψ n = √ Pr n e iθ n where θ n is a phase factor. But Tucci does not provide a solution for estimating the phase parameters. By considering different values for the phase factor, every CBN can be represented as an unlimited number of QBNs. Here, we present an example of QBN for the PD task. As shown in 11 , the probability of choosing defection by the second prisoner (B) is estimated by applying the quantum counterpart of Eqs. (1)(2)(3) and Born rule 52 in QIT.
where α is a normalizations factor which is obtained by (10): Similar to considering wave-particle duality in Young's two-slit experiment, the wave behavior is removed in certain situations. However, there may be destructive or constructive interferences under uncertainty if at least two wave functions interact with each other. There were different attempts to describe Young's two-slit experiment by non-Kolmogorovean probabilistic model such as proposing the p-adic theory of probability 54 . Inspired by this experiment, Khrennikov 55 introduces the measure of statistical perturbations and considers interference terms in total probability law. He proposes the Quantum-like Representation Algorithm (QLRA) and modifies the classical TPL by assuming a trigonometric or hyperbolic interference term 56 . Considering the interference effects in TPL is a key idea to justify the contradictions in most quantum-like decision models. Including this effect is also the most important challenge of these models to provide predictions of human selections.
Most quantum-like decision theories only justify the observations by selecting suitable interference-effect. While implementing the predictive decision theory needs to estimate the interference term instead. Quantum decision theory (QDT) 57 is one of the first predictive approaches for estimating the interference term. This model takes into account objective utility related to expected benefit, as well as subjective interference. By applying some quantum parameters, this model assumes the interference term at a static value of 0.25. Wang et al. 25 show that Bayesian and Markov models cannot account for empirical data on human behavior. In addition to justifying empirical data of human selections by considering a static value of interference term for each scenario, they present a predictive QQ model which is supported by empirical tests of order effects. After that, Moreira et al. 14 added a predictive feature to the QBN structure using a piecewise heuristic function based on probabilities in the equivalent CBN. In 2019, Huang et al. 58 presented a solution based on the belief entropy concept. They use Deng entropy 59 to estimate the interference term, which is a generalized form of Shannon entropy and is defined as 59 : (5) |ψ� = c 1 e iθ 1 |0� + c 2 e iθ 2 |1� = c 1 e iθ 1 c 2 e iθ 2 T (6) Pr(|i�) = |P i |ψ�| 2 = �ψ|P i |ψ�.   www.nature.com/scientificreports/ where m is a mass function defined on X , A is the focal element of m and |A| is equal to the cardinality of A 59 . Their model has some drawbacks leading to an estimated value of more than one for the cosine function in the interference term 27 . So, according to probability values in certain situations presented in BN tables, it is possible to achieve the unadmitted complex number for probability in some cases 27 . Another heuristics approach is proposed by Dia et al. 60 . In this method, a coefficient r , based on wave functions in QBN, is defined to estimate the interference term. At the same time, Wichert et al. 61 propose the Balanced Quantum-Like Bayesian Networks. They introduce the law of balance based on the notion of balanced intensity waves and the law of maximum uncertainty based on the entropy concept. The main idea of this method is to balance the intensity waves resulting from quantum interference in such a way that, during Bayes normalization, they cancel each other. Finally, a predictive entangled QBN (PEQBN) structure is introduced in our previous study 27 , motivated by the QIT instead of heuristics approaches. Entangled nodes in the PEQBN model mean that each node/DM is considered a part of the whole system. Therefore, the effect of indirect relations that cannot be modeled as classical arcs are estimated by two entanglement measures in QIT, and the proposed qW witness 27 .

Method
Here we present a biased variant of entangled quantum-like BN from a cognitive perspective to examine bias behavior in the decision-making process. We consider different bias behaviors due to the emotions between agents or past personal experiences motivated by the entanglement concept. Besides, the effect of unequal probabilities in the parent node, obtained from past experiences of other agents, is modeled that is inspired by electric fields. We simulate a two-step binary decision-making experiment in an entangled quantum-like BN structure 27 and predict the probabilities in the child nodes without any observations of parent nodes. The dynamical evolution of selecting each choice under uncertainty is simulated by overlapped wave functions 12 that cause an interference effect because of the existence of phase parameters. The problem of justifying the phase parameter from the perspective of social systems is the most important unsolved challenge of quantum-like models. Providing an acceptable justification for these parameters can pave the way to estimating the interference terms and proposing predictive quantum-like models. The phase parameters in the superposition state of a quantum system are related to the environmental condition and obtained from the initial experimental setting in a physical system. Although these parameters are unknown in social systems, they are attributed to the contextual conditions of the DM 33 , the correlation between the phenomena 42 , or the degree of uncertainty of the DM 62 to the decision scenario, in different quantum-like models. In this study, we attribute the phases in the superposition state to the initial mental conditions of the decision-maker, which is a function of personal experiences, spiritual conditions, and the decision-maker's relations with the other agents in the society. So we use these parameters to mode bias behavior in the human mind and unknown relationships between agents. In this regard, the introduced social entanglement concept, inspired by quantum entanglement in our previous study 27 , has a key role in our method. This means that although we are not dealing with a quantum system, the concept of quantum-like entanglement has been borrowed from quantum physics due to the similarity between entangled quantum systems and social systems. Similar to entangled quantum systems, information about the social system is not complete and the information is shared in the whole system. Also, as it is not possible to separate the information of each particle in an entangled quantum system, decision-makers in a social system are not considered isolated agents. Unknown relations between agents influence the decision-making process, and so it is impossible to measure the initial mental conditions of each decision-maker separately.
On the other hand, as quantum interference is related to entanglement in quantum physics, we assume that the interference term in the social system is related to the social entanglement. The fundamental differences between quantum and classical interference arise from non-separability in quantum systems which leads to interference between probability amplitudes rather than between physically existing realistic objects, like the electromagnetic waves 57,63 . There are several common properties between decision-making processes and quantum interference. The intention interference and non-commutativity of subsequent decisions are critical characteristics of human decision-making. The non-commutativity of actions (in the framework of quantum operators) and entanglement are essential properties of quantum interference. In contrast, classical interference cannot include these effects in the system due to its commutative nature and locality. So, as assumed in our previous PEQBN model, we relate the interference term to the entanglement between agents 27 E(ρ) : where ρ = |ψ >< ψ| . In QIT, entanglement considers long-range effects because of uncertainty sources in the dynamical behavior of physical system. Although there is no exact tool for measuring the entanglement of a quantum composite system, there are many criteria, known as quantum measures, to estimate entanglement values or at least find the boundaries of this parameter. In addition, there are necessary and sufficient entanglement criteria in terms of directly measurable observables, called entanglement witnesses 64 . Because entanglement witnesses are directly measurable quantities, they are very useful for the experimental analysis of entanglement 64 . So, because the initial conditions of social systems are not well known, we cannot estimate entanglement measures directly, and we introduce a measurable function as a quantum-like witness of social entanglement. Then we obtain its relations to the concurrence entanglement measure, based on empirical data, to estimate entanglement in the social system and hence the interference term. www.nature.com/scientificreports/

Modeling the bias behavior inspired by the entanglement concept. This section introduces a new
quantum-like entangled witness in Hilbert space for modeling bias behavior caused by emotions such as friendship or enmity between agents or past personal experiences from a cognitive viewpoint. It is noted that bias is usually interpreted as prior information in Bayes' formula. Still, this interpretation only considers the effect of bias behavior on the amplitude of wave function in a QBN structure. In this study, we consider the effect of bias behavior on phase parameters too. So, we assume that prior information such as past experiences or relationships with other agents can change the decision-maker's initial state, including amplitude and phase parameters. The above Hilbert space represents all possible results after a measurement. So until we are not informed about the results of a decision-making process (doing a measurement) in nodes A and B , the superposition state is formed as follows: where c i and θ i are defined in Table 2. The first step for estimating the Pr(B = b 1 ) without measuring A is applying the projector P B=b 1 = |b 1 ��b 1 | on the superposition state |ψ�. This equation is obtained by applying Eq. (6) on the wave function |ψ� . The third term in Eq. (15) is related to the interference effect, which adds a stochastic long-range impact to the problem. If we remove the uncertainty from the first node by doing an observation, the third term becomes equal to zero. So, Pr(B = b 1 ) becomes equal to c 1 2 or c 3 2 according to the result of measuring A . Now let us consider the unknown situations in which there is no information about node A . In this case, we need to estimate the interference term related to the initial condition in the superposition state/initial bias in mind. The key parameter of the interference term is cos(θ) , which is a function of initial phase parameters, used for modeling bias behavior in this study. So, inspired by physical systems, we relate the interference term to the initial experimental situation. But in social system, initial experimental situations are the function of experiences of DM or unknown relations between agents in the society. Inspired by the entanglement concept in quantum physics, which models the unknown relation between Pr(B = b 1 ) = P B=b 1 |ψ� 2 = �ψ P B=b 1 ψ� = (|b 1 ��b 1 |ψ�) † (|b 1 ��b 1 |ψ�) = |�b 1 |a 1 ��a 1 |ψ�| 2 + |�b 1 |a 2 ��a 2 |ψ�| 2 + (|�b 1 |a 1 �||�a 1 |ψ�||�ψ|a 2 �||�a 2 |b 1 �|cos θ) = c 2 1 + c 2 3 + (|�b 1 |a 1 �||�a 1 |ψ�||�ψ|a 2 �||�a 2 |b 1 �|cos θ ).
A B 1 Figure 2. An entangled bayesian network including two nodes. Each node (A and B) is considered a part of the whole system (illustrated by a yellow oval). The nodes in the social system are entangled due to unknown sources such as trust and cooperation between agents. In the BEQBN, probabilities in BNs are replaced by suitable complex wave functions. Table 2. Wave functions in the BEQBN model presented in Fig. 2. www.nature.com/scientificreports/ particles in a whole system, we introduce a similar concept called social entanglement in a quantum-like multiagent system that is only partially known. So, here we simulate the social system as a composite quantum system. Then we attribute the interference effects to Shannon entropy as a measure of entanglement. Based on quantum studies Shannon entropy ( E Sh ) is related to the concurrence (C) , a common measure of quantum entanglement, as shown below: Because the initial conditions of a social system and density matrix ( ρ ) are not well known, we cannot estimate E(ρ) directly, so an observable function called quantum-like entanglement witness (qlw) is introduced in situations with certainty. By finding the relation between qlw and (C) based on empirical data, we estimate social entanglement in the uncertain case. Afterward, Shannon entropy and the interference term (cos (θ)) are estimated by measuring the system entanglement in a practical way.
Let us review this approach in more details. For estimating this parameter that is inspired by the entanglement concept, a novel quantum-like witness in Hilbert space is introduced in this study. Unlike previous methods, we use a quantum solution in a four-dimensional Hilbert space based on QP for solving a quantum-like problem. In the QBN structure, such as shown in Fig. 2, regardless of the condition in the first node, the uncertainty in the second node is eliminated by observing node B.
According to QIT, the operator I ⊗ P B=b 1 tells us to leave the first node unobserved and apply the P B=b 1 operator to the second node 52 . Here we define two input vectors v 1 , and v 2 as follows: These input vectors are defined in four-dimensional Hilbert space by applying a quantum view to the problem instead of the classical view used to define two-dimensional vectors in other QBN models. Finally, an observable function called quantum-like witness ( qlw ) is suggested based on these vectors as shown below: For estimating the interference term, a relationship between the introduced qlw witness and C is obtained by applying the S R TLBO optimization algorithm 65 . This relationship is defined in a way that optimizes the concurrence as much as possible to achieve the best fit between predicted and experimental probabilities. Applying optimization in this procedure is meaningful because optimization is used in the definition of many quantum entanglement measures and witnesses due to some reasons such as considering the best initial conditions 66 . So we use the following equation to estimate the concurrence entanglement measure: where n is equal to 2 4 . This equation guarantees that the C value is obtained between 0 and 1, which is compatible with the acceptable range for this entanglement measure. By applying Eq. (16), Shannon entropy, as a measure of entanglement, is estimated. After that, cos(θ) in the interference term is obtained by Eq. (12), inspired by QIT.

Modeling the bias behavior inspired by electric fields.
The main advantage of this study is modeling the mind's initial bias in the decision-making process due to unequal probabilities in the parent node obtained from past decisions of other agents in society. In most studies on QBN structures, the quantum-like approaches are evaluated on simple two-step binary decision-making tasks, with equal probability in the parent node, such as prisoner's dilemma and two-step gambling games 67 . Let us focus on situations in which Pr(a1) ∼= Pr(a2 ) in the first node. When a decision between two choices has not been made yet, we think about both options simultaneously, and a superposition state is formed in the QBN structure. The coefficients of each state in the superposition are complex numbers. Unequal probabilities affect the amplitude of these coefficients, but we also consider the effect of unequal probabilities on the initial phase. For modeling this effect, we are inspired by the deviation of the proton during the passage between two plates. Imagine some electrons being transferred from one plate to another. In this case, the percentage of positive charges of each plate changes from 50%, and an electric field is created between two charged plates, which diverts the passing proton to one of the plates (Fig. 3a). Similarly, imagine that the probability of one option in a binary problem is more than 50%, despite being unaware of the results, the initial condition of our mind tends toward the option with a higher probability. In this regard, we propose a hypothetical bias potential field to model this bias behavior on initial phase parameters (Fig. 3b).
Classical models consider that DMs think about each option separately (Fig. 4a), even in the presence of uncertainty in node A (parent node). The proposed method considers this situation, only if uncertainty is (19) C = 0.136 − 0.03 cos n qlw + 0.02 * sin n qlw − 0.029 * cos 2n qlw − 0.12 * sin 2n qlw . www.nature.com/scientificreports/ removed by an observation. While from a quantum-like viewpoint, node A can exist in the following superposition state (Fig. 4b) under uncertainty: where κ 2 = 1 − κ 1 2 . The effect of unequal probabilities has appeared in the amplitude parameters (κ i ) . However, phase parameters (γ i ) and, therefore, interference terms are also affected, indirectly. When a DM wants to choose an option in the second node that can be beneficial or detrimental, both options with unequal probabilities in the first node are considered simultaneously (Fig. 4b).
Here, we suggest that DM's mind tends to the event that is more likely to happen. Since we attribute the initial phases to the DM's mental conditions, we need to model the effect of this bias on (γ i ) parameters and, therefore, the interference term. We model the bias effect to initial phases by a hypothetical field motivated by the behavior of particles in an electric field (Fig. 4c). So most DMs/qubits tend to have a particular bias. In this regard, a potential function U(κ 1 ) is defined to model bias behavior (Fig. 5), and the supplementary term shown in Eq. (21) is added to Eq. (15): If two options have equal probabilities, no significant bias is observed and U √ 0.5 ≈ 0 . Also, U(0) = U(1) ≈ 0 because there is no uncertainty in these situations. But if the probability of choosing one option is 0.75 and the other one is 0.25, U(κ 1 ) is maximized. The proposed bias function (Fig. 5) is symmetric because the probability of one option Pr(A = a 1 ) is equivalent to the probability value of (1 − Pr(A = a 1 )) for the second option. So, we can predict Pr(B = b 1 ) without measuring A as follows: (20) |ψ A � = κ 1 e iγ 1 |a 1 � + κ 2 e iγ 2 |a 2 � = κ 1 e iγ 1 κ 2 e iγ 2 T .  www.nature.com/scientificreports/ The pseudo-code of the proposed BEQBN method is presented in Table 3.

Results
Here, the proposed BEQBN is evaluated on two decision-making scenarios with equal and unequal probabilities in the first node. To provide a fair comparison, Obtained results by BEQBN compare to the estimated value by CBN and six quantum-like decision-making models.
Evaluation of the proposed BEQBN on PD game. The PD game is a benchmark in the decisionmaking domain with equal probabilities in the first node, described in more detail in the preliminaries section. We apply the proposed model, CBN, and six other predictive quantum-like methods, including QDT 57 , QBN 14 , QBN 58 , QBN 60 , BQBN 61 , and PEQBN 27 , to different versions of the PD game. This evaluation is based on the empirical results reported in Table 1. The predicted probability for choosing the defect option by the second player is estimated by each method. The errors between predicted and empirical probabilities for these (22) Table 3. The Pseudo-code of the BEQBN model for a binary decision-making problem.
Step 1: Two vectors for states of each binary node are presented as |a 1 � = 1 0 T and |a 2 � = 0 1 T in Hilbert space H 1 and |b 1 � = 1 0 T and |b 2 � = 0 1 T in H 2 Step 2: The composite Hilbert space H = H 1 ⊗ H 2 and four basis vectors are obtained by applying the tensor product Step 3: Superposition model |ψ� is formed as presented in Eq. (14) Step 4: Two input vectors v 1 , and v 2 , are obtained by Eq. (17) Step 5: Quantum-like witness ( qlw ) is calculated by Eq. (18) Step 6: The concurrence entanglement measure ( C ) is calculated by Eq. (19) Step 7: Shannon entropy E Sh is calculated by Eq. (16) Step 8: The biased operator U is estimated by Eq. (21) Step 9: The projector P B=1 is applied on |ψ� , and cos(θ) in Eq. (15) is considered equal to −E Sh .
Step 10: Pr(B = b 1 ) is estimated by Eq. (22)  www.nature.com/scientificreports/ approaches are reported in Table 4. Also, these results along with their root mean square error ( RMSE ) values are plotted in Fig. 6. According to this evaluation, the predictions by the proposed BEQBN model are in good agreement with the empirical data. Due to considering bias behavior, BEQBN achieves the first rank in this evaluation by obtaining an RMSE value equal to 3.4. Because of the equal probability of defection or cooperation of the first prisoner, mental bias modeled by bias potential function is neglected in this task. However, mental bias due to emotions between two prisoners is modeled by the qlw witness and two entanglement measures. This effect is justified by the previous knowledge of the two prisoners of moral character and cooperation between them, although they have been in separate cells. Our previous entangled BN 27 and the heuristic method presented by Moreira et al. 14 , rank second to third in this evaluation, respectively. Other compared approaches, including QBN 58 , QBN 60 , QDT 57 , BQBN 61 , and classical BN, are ranked fourth to eighth.

Evaluation of BEQBN on face categorization and decision.
In the second evaluation, the proposed model is applied to a two-step scenario with unequal probabilities in the first node, while selection in the second node has benefit or loss for the DM. This task is presented by Busemeyer et al. 68 about categorizing people based on their faces. In this task, there are two separated conditions: (1) categorization before decision-making (C-D), and (2) decision-making (D alone).
Same pictures of faces are shown to participants in both situations that varied in two dimensions: face width and lip thickness. Participants in the C-D condition are asked to classify the face as either a "good" or "bad" and then choose whether to "attack" or "withdraw." Faces are classified into two groups: "narrow faces" with a narrow breadth and thick lips and "wide faces" with a wide breadth and thin lips. The participants are informed that "narrow" faces have a 0.83 chance of being in the "bad guy" population, whereas "wide" faces have a 0.84 probability of being in the "good guy" population. Attacking the "bad man" and retreating from the "good guy" are rewarded. The participants in the D alone condition are instructed to decide without any classifications.
We apply the proposed BEQBN and seven other predictive methods, including CBN, QDT 57 , QBN 14 , QBN 58 , QBN 60 , BQBN 61 , and PEQBN 27 , to this scenario and predict the probability of choosing "attack" in the D alone condition based on information presented in Table 5. According to our approach, categorization in the first round changes the initial conditions of the DM's mind and therefore the superposition state of each DM. So categorization and decision-making in this task are two entangled processes. In D alone scenario, participants think about categorization as good or bad guys simultaneously. The interference term is estimated by entanglement measures and caused by changing the initial state of the mind in the categorization process. Another human bias is also considered, due to the unequal probabilities of whether each face is good or bad in the first node.  (Table 1) and predicted values of Pr(B = D) . The results are obtained by applying seven models, including CBN, six recent quantum-like decision models, and the proposed BEQBN model in this study to the PD game.  www.nature.com/scientificreports/ Decision-makers think simultaneously about two options, but they are more inclined towards one of them. This bias is modeled by adding the potential function U introduced in Eq. (21) As shown in Table 6, this correction reduces the error between predicted and empirical results. In this table, BEQBN wins the first rank by achieving the RMSE value equal to 5.07. The results obtained by other compared methods are shown in Table 6 and Fig. 7. According to these results, the other six quantum-like models are no more successful than classical Bayesian network structure, and CBN, QBN 60 , QBN 58 , QBN 14 , PEQBN 27 , QDT 57 , and BQBN 61 are ranked second to eighth, respectively. Therefore, in the situations where we deal with unequal probabilities in the first node, considering only one aspect of interference does not improve the performance of BN structures.

Conclusion
Future autonomous systems will need cognitive capabilities, including reasoning and decision making. Hence, modeling human selection behavior is one of the grand challenges in the future of control and systems. This work extends quantum-like approaches to modeling human choice behavior by focusing on human biases from a cognitive viewpoint. The proposed model is built upon the basis of the entangled quantum-like BN due to the synergy of three powerful approaches, including (1) classical BN for modeling causal relationships, (2) quantum probability for adding a new dimension for modeling the states of the human mind in the complex domain, and (3) entanglement property for modeling the long-range effects due to uncertainty sources in the dynamical behavior of the decision-making process. By simulating a two-step binary choice task as a composite entangled quantum system, we consider each node of a BN structure as a particle or wave in certain and uncertain situations, respectively.
The point of departure in this study is modeling the initial bias behavior of the DM's mind due to unequal probabilities in the unmeasured parent node of the BN structure. To this end, we define a potential function inspired by an electric field to estimate the effect of bias behaviors on the interference effect under uncertainty. Also, a novel quantum-like witness in the Hilbert space is introduced to model bias behavior due to emotions between agents in the social systems and past personal experiences. By applying the proposed quantum-like witness, a quantum solution in Hilbert space rather than classical ones in other models is used to solve a quantum-like problem. We also present a periodic relationship between the proposed quantum-like witness and the well-known concurrence entanglement measure in QIT. Then Shannon entropy is estimated based on the relations in quantum information theory and applied for modeling the destructive or constructive interference effects between the two choices in a binary task under uncertainty.
Finally, the proposed BEQBN is evaluated successfully on two decision-making scenarios with equal and unequal probabilities in the first node. According to experimental results, BEQBN predicts the probability of selecting each option under uncertainty with minimum error compared to classical BN and six recent quantumlike Bayesian networks. The results indicate that the proposed model in this study archives the first rank in overall evaluations by considering mind's biased behaviors.
In the future, we hope to extend our approach for modeling more complex situations with three different options for each step. For this purpose, we also plan to design new tasks for gathering empirical data and creating new databases. Then, we can evaluate the application of the proposed model in solving human challenges such as crisis management, energy management, or traffic forecasting.

Data availability
All data generated or analyzed during this study are included in this published article [and its supplementary information files].  Figure 7. Graphical illustration of errors between empirical results (Table 5) and predicted probability of taking defensive action without categorization. Predicted results are obtained by applying CBN, six quantum-like decision models, and the proposed BEQBN model in this study on Categorization-Decision Experiment 68 .